If P = `{2 m : m in N} and Q = {2^(m) : m in N}` , where m is positive, then :

Options

A. `Q sub P`
B. `P sub Q`
C. P = Q
D. `P cup Q = N`

Answer

To solve the problem, we need to analyze the sets P and Q given in the question.

### Step 1: Define the sets P and Q

The set P is defined as:
\[ P = \{ 2m : m \in \mathbb{N} \} \]
This means that P consists of all numbers that can be expressed as \( 2 \times m \), where \( m \) is a natural number (1, 2, 3, ...).

The set Q is defined as:
\[ Q = \{ 2^m : m \in \mathbb{N} \} \]
This means that Q consists of all numbers that can be expressed as \( 2 \) raised to the power of \( m \), where \( m \) is a natural number.

### Step 2: List the elements of set P

Let's calculate the first few elements of set P by substituting values of \( m \):
- For \( m = 1 \): \( 2 \times 1 = 2 \)
- For \( m = 2 \): \( 2 \times 2 = 4 \)
- For \( m = 3 \): \( 2 \times 3 = 6 \)
- For \( m = 4 \): \( 2 \times 4 = 8 \)
- For \( m = 5 \): \( 2 \times 5 = 10 \)

So, the set P can be written as:
\[ P = \{ 2, 4, 6, 8, 10, ... \} \]

### Step 3: List the elements of set Q

Now, let's calculate the first few elements of set Q:
- For \( m = 1 \): \( 2^1 = 2 \)
- For \( m = 2 \): \( 2^2 = 4 \)
- For \( m = 3 \): \( 2^3 = 8 \)
- For \( m = 4 \): \( 2^4 = 16 \)
- For \( m = 5 \): \( 2^5 = 32 \)

So, the set Q can be written as:
\[ Q = \{ 2, 4, 8, 16, 32, ... \} \]

### Step 4: Compare the two sets P and Q

Now we compare the elements of both sets:
- Set P contains: \( 2, 4, 6, 8, 10, ... \)
- Set Q contains: \( 2, 4, 8, 16, 32, ... \)

### Step 5: Determine the relationship between P and Q

From our lists:
- The elements \( 2 \) and \( 4 \) are common in both sets.
- Set P contains additional elements like \( 6, 10, ... \) which are not in set Q.
- Set Q contains elements like \( 8, 16, 32, ... \) which are not in set P.

Thus, we can conclude:
- Set P is not a subset of Q because it contains elements (like 6, 10) that are not in Q.
- Set Q is not a subset of P because it contains elements (like 8, 16) that are not in P.
- Therefore, neither set is equal to the other.

### Final Conclusion

The correct option is:
- **Option A**: Q is a subset of P is incorrect.
- **Option B**: P is a subset of Q is incorrect.
- **Option C**: P is equal to Q is incorrect.
- **Option D**: P subset of Q is equal to N is incorrect.

Since none of the options are correct, we conclude that the relationship between P and Q is that they are distinct sets with some common elements.